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Risky Arbitrage, Asset Prices, and Externalities
Cuong Le Van, Frank H. Page, Myrna H. Wooders
To cite this version:
Cuong Le Van, Frank H. Page, Myrna H. Wooders. Risky Arbitrage, Asset Prices, and Externalities. Economic Theory, Springer Verlag, 2007, 33 (3), pp.475-491. �10.1007/s00199-006-0151-1�. �halshs-00102698�
Risky Arbitrage, Asset Prices, and Externalities
Cuong Le Van CES and CNRS, University of Paris 1 75013 Paris France levan@univ-paris1.fr Frank H. Page, Jr. Department of Finance University of Alabama Tuscaloosa, AL 35487 USA fpage@cba.ua.edu Myrna Wooders∗ Department of Economics Vanderbilt University Nashville, TN 37235 USA m.wooders@vanderbilt.eduCurrent Version, June 2006†
Abstract
We introduce a no-risky-arbitrage price condition (NRAP) for asset market models allowing both unbounded short sales and externalities such as trading volume. We then demonstrate that NRAP is suffi-cient for the existence of competitive equilibrium in the presence of externalities. Moreover, we show that if all risky arbitrages are utility increasing, then NRAP characterizes competitive equilibrium in the presence of externalities.
JEL classifications: C62, D50
KEYWORDS: Risky Arbitrage, Competitive Equilibrium, Viable Asset Prices
∗Also, Department of Economics, University of Warwick, Coventry CV4 7AL, UK †We are indebted to an anonymous referee for careful reading and helpful comments on
an earlier version of this paper. Page and Wooders are especially grateful to CERMSEM and EUREQua for their support and hospitality which made possible our collaboration.
1
Introduction
In competitive asset markets trading volume influences investors’ expecta-tions of future asset returns, and thus, influences equilibrium asset prices. The influence of trading externalities, such as trading volume on equilib-rium asset prices, is brought about by a process of arbitrage elimination that characterizes informationally efficient asset markets. While there have been numerous papers investigating the connections between arbitrage and equilibrium asset prices in asset market models with unbounded short sales, with one exception there has been no work on the connections between arbi-trage and asset prices in models with short sales where trading externalities are taken into account.1
In this paper, we introduce a no-risky-arbitrage price condition, NRAP, for models allowing both trading externalities and unbounded short sales, and demonstrate that NRAP is sufficient, and in some cases necessary, for the existence of competitive equilibrium in the presence of externalities. In empirical studies of financial markets, available information may well include both prices and volumes of net trades. Thus, it is important to have characterizations depending on prices and observable data. In fact, our research follows the fundamental work of Hammond (1983) for asset market models and Werner (1987) for general equilibrium models.
In a risky arbitrage, an agent sells an existing portfolio and buys a utility nondecreasing alternative portfolio for a net cost less than or equal to zero. Whether a particular pair of transactions (selling a portfolio and buying an-other) constitutes a risky arbitrage thus depends on the agent’s preferences as well as asset prices and, in the presence of externalities, each agent’s pref-erences in turn depend directly on the trades of other agents. In its most potent form, a risky arbitrage is utility increasing and generates a net cost less than or equal to zero.2 Here, we formalize the notion of risky arbitrage in an asset market model with trading externalities and short sales and intro-duce a condition on asset prices that rules out risky arbitrage for all agents. Given the close connection between agent preferences and risky arbitrage, NRAP is essentially an assumption concerning the degree of homogeneity in
1See Le Van, Page, and Wooders (2001).
2In a riskless arbitrage, an agent sells an existing portfolio and buys a replicating
portfolio (i.e., an alternative portfolio having the same returns in all states of nature) for a net cost less than or equal to zero. Thus, a riskless arbitrage is a special case of a risky arbitrage. In its most potent form, a riskless arbitrage generates a positive amount of money upfront - or put differently, a riskless arbitrage can be carried out via a pair of trades having a net cost strictly less than zero.
agents’ preferences.
The intuition behind our results is straightforward: with sufficient homo-geneity, even if trading externalities are present and unbounded short sales are allowed, if NRAP is satisfied an agent will be unable to carry out a risky arbitrage because no one will be willing to take the other side of the trans-action. However, with externalities, carrying out a transaction may perturb the arbitrage opportunities for all agents and lead to further changes, even reversing the desirability of the initial transaction. Such considerations make formulation of NRAP delicate.
Besides being sufficient for existence of equilibrium, whenever all risky arbitrages are utility increasing then NRAP is also necessary for existence of equilibrium. Thus, in asset markets with externalities and short sales in which all risky arbitrages are utility increasing, NRAP characterizes compet-itive equilibrium. Moreover, for any given level of the externalities, NRAP ensures existence of demand functions.
In the literature, no-risky-arbitrage (NRA) conditions for asset market models without trading externalities fall into three broad categories: (i) con-ditions on net trades, for example, Hart (1974), Page (1987), Nielsen (1989), Page, Wooders, and Monteiro (2000), and Allouch (2002); (ii) conditions on prices, for example, Grandmont (1970,1977), Green (1973), Hammond (1983), and Werner (1987); (iii) conditions on the set of utility possibilities (namely compactness), for example Brown and Werner (1995) and Dana, Le Van, and Magnien (1999). In Le Van, Page, and Wooders (2001) an NRA condition on net trades is introduced for models with trading externalities and short sales - a condition that reduces to the condition of Hart (1974) if no externalities are present - and it is shown that the net trades NRA condition is sufficient for existence. Since NRAP reduces to the condition of Werner (1987) if no externalities are present and enables proof of existence of equilibrium in the presence of externalities, our research continues the prior work. We further relate our condition to prior conditions by show-ing that, if all risky arbitrages are utility increasshow-ing, then NRAP and the NRA net trades condition are equivalent, and both characterize competitive equilibrium.
In an economic model similar to the model presented here, but without externalities, Dana, Le Van, and Magnien (1999) have shown that com-pactness of the set of utility possibilities is sufficient for the existence of competitive equilibrium. However, in the presence of externalities compact-ness of utility possibilities, as a condition limiting arbitrage, is not sufficient for existence.
2
An Economy with Trading Externalities
Let (Xj, ωj, uj(·, ·))nj=1 denote an unbounded exchange economy with
trad-ing externalities. In the economy (Xj, ωj, uj(·, ·))nj=1each agent j has choice
set Xj ⊂ RLand endowment ωj ∈ Xj.The jth agent’s preferences, defined
over X :=Qnj=1Xj,are specified via a utility function uj(·, ·) : Xj× X−j →
R, where X−j :=Qi6=jXi. Note that for all agents j, X = Xj × X−j. Let
x−j denote a typical element of X−j. Often it will useful to denote the
elements in X by (xj, x−j).
The set of rational allocations is given by
A= {(x1, ..., xn) ∈ X :Pj=1n xj =Pnj=1ωj
andf oreachj, uj(xj, x−j) ≥ uj(ωj, x−j)}.
(1) We will denote by A−j the projection of A onto X−j.
For each (xj, x−j) ∈Qnj=1Xj,the preferred set is given by
Pj(xj, x−j) := {x ∈ Xj : uj(x, x−j) > uj(xj, x−j)}, (2)
while the weakly preferred set is given by
b
Pj(xj, x−j) := {x ∈ Xj : uj(x, x−j) ≥ uj(xj, x−j)}. (3)
We will maintain the following assumptions on the economy (Xj, ωj, uj(·, ·))nj=1:
For each j = 1, ..., n, [A − 1]
(
Xjisclosedandconvex, andωj ∈ intXj,
where“int” denotes“interior′′.
[A − 2]
(
F oreach(xj, x−j) ∈ X, uj(·, x−j)isquasi − concaveonXj,
anduj(·, ·)iscontinuousonXj× X−j.
[A − 3]
(
F oreach(xj, x−j) ∈ A, Pj(xj, x−j) 6= ∅,
andclPj(xj, x−j) =Pbj(xj, x−j).
Note that in [A-1] we do not assume that consumption sets, Xj, are bounded.
Also, note that given [A-2], for all (xj, x−j) ∈ X the preferred set Pj(xj, x−j)
is nonempty and convex, while the weakly preferred setPbj(xj, x−j) is
non-empty, closed and convex. Finally, note that [A-3] implies that there is local nonsatiation at rational allocations.
Given prices p ∈ RL, the cost of a consumption vector x = (x
1, ..., xL) is
hp, xi =PLℓ=1pℓ· xℓ.The budget set is given by3
Bj(p, ωj) = {x ∈ Xj : hp, xi ≤ hp, ωji}. (4)
Without loss of generality we can assume that prices are contained in the unit ball
B := {p ∈ RL: kpk ≤ 1}.
An equilibrium for the economy (Xj, ωj, uj(·))nj=1 is an (n + 1)-tuple of
vectors (x1, ..., xn, p) such that
(i) (x1, ..., xn) ∈ A (the allocation is feasible);
(ii) p ∈ B\{0} (prices are in the unit ball and not all prices are zero); and
(iii) for each j,
(a) hp, xji = hp, ωji (budget constraints are satisfied), and
(b) xj ∈ Bj(p, ωj) and Pj(xj, x−j) ∩ Bj(p, ωj) = ∅ (i.e., xj maximizes
uj(xj, x−j) over Bj(p, ωj)).
We provide an example illustrating our model in application to an asset market. This example will be further developed later in the paper.
Example: Part 1, An Asset Market with Trading Externalities. Consider an agent j who seeks to form a portfolio xj = (x1j, . . . , xLj) of
Lrisky assets so as to maximize his expected utility given by uj(xj, x−j) =
Z
RL
Uj(hxj, ri)dµj(r|x−j).
Here, xij denotes the number of (perfectly divisible) shares of asset i held
in the jth agent’s portfolio xj, and ri denotes the return on asset i, i.e., the
ith component of the asset return vector r ∈ RL
+.4 The inner product of the
portfolio vector xj and the asset return vector r, denoted by
hxj, ri = L
X
i=1
xijri,
3The restriction of the budget set to be a subset of the consumption set entails no losss
of substance or generality.
4RL
+denotes the nonnegative orthant of RL. Thus, here we are assuming that all asset
gives the return generated by portfolio xj if the realized asset return vector
is r. Note that because short sales are allowed, hxj, ri can be negative. The
function
Uj(·) : R → R
is the jth agent’s utility function defined over end-of-period wealth. The
probability measure µj(·|x−j) defined over Borel subsets of asset returns
represents the jth agent’s subjective probability beliefs concerning end-of-period asset returns conditioned by the (n − 1)-tuple, x−j, of portfolios held
by other agents.
Denote by S[µj(·|x−j)] the support of the conditional probability
mea-sure µj(·|x−j), and by K(x−j) the smallest convex cone containing S[µj(·|x−j)].
Finally, let K+(x
−j) denote the positive dual cone of K(x−j), that is, let
K+(x−j) :=
n
y∈ RL: hy, ri ≥ 0∀r ∈ K(x−j)
o
.
Note that any vector of net trades y contained in K+(x−j) generates a
nonnegative return with probability 1. Thus, trading in any direction y ∈ K+(x−j) is without downside risk.
Assume the following:
(a-1) For each agent j = 1, 2, . . . , n, the utility function Uj(·) : R → R is
concave and increasing.
(a-2) For each agent j = 1, 2, . . . , n, the mapping, x−j → µj(·|x−j),
is continuous in the topology of weak (or narrow) convergence of prob-ability measures.
(a-3) For all rational allocations (xj, x−j) ∈ A and for all agents j =
1, 2, . . . , n, S[µj(·|x−j)] ∩ RL+\{0} 6= ∅.
(a-4) For all x−j ∈ X−j and for all agents j = 1, 2, . . . , n,
S[µj(·|x−j)] ⊆ C
for some bounded set C ⊂ RL +.
(a-5) For all agents j = 1, 2, . . . , n, the portfolio choice set Xj is closed
and convex with initial portfolio ωj ∈ intXj, and for all (xj,x−j) ∈ X,
y∈ K+(x
In words, assumption (a-3) means that at rational allocations each agent believes that some asset will generate a positive return with a positive prob-ability. Assumption (a-5) means that given any configuration of starting portfolios (xj,x−j) ∈ X, agent j can alter (or rebalance) his starting
port-folio xj via net trades y ∈ K+(x−j) (i.e., via a no-downside-risk portfolio)
without violating portfolio feasibility (i.e., without violating his constraint set Xj). Note that together assumptions (a-1), (a-3), and (a-5) imply that
agents’ expected utility preferences satisfy assumption [A-3] (local nonsatia-tion) while assumptions (a-1) and (a-2) imply that agents’ expected utility preferences satisfy assumptions [A-2] (quasiconcavity and continuity).
3
Risky Arbitrage and NRAP
We begin by recalling a few basic facts about recession cones (see Section 8 in Rockafellar (1970)). Let X be a convex set in RL.The recession cone
0+(X) corresponding to X is given by
0+(X) = {y ∈ RL: x + λy ∈ X f orallλ ≥ 0andx ∈ X}. (5)
If X is also closed, then the set 0+(X) is a closed convex cone containing
the origin. Moreover, if X is closed, then x + λy ∈ X for some x ∈ X and all λ ≥ 0 implies that x′+ λy ∈ X for all x′ ∈ X and all λ ≥ 0. Thus, if X
is closed, then we can conclude that y ∈ 0+(X) if for some x ∈ X and all
λ≥ 0, x + λy ∈ X. (Risky Arbitrage):
A vector of net trades yj ∈ RL is a risky arbitrage for agent j if there
exists a sequence n xko k= n (xkj, xk−j)o k = n (xk1, . . . , xkn)o k⊂ X such that f orallk, uj(xkj, xk−j) ≥ uj(ωj, xk−j), and yj = limktkxkj f orsomesequencentko kof positiverealnumbers suchthattk↓ 0.
Denote by Rj the set of all risky arbitrages for agent j.
Let (xj, x−j) ∈ X satisfy xj ∈Pbj(ωj, x−j). If yj ∈ 0+Pbj(ωj, x−j), then yj
cone of the weakly preferred setPbj(ωj, x−j) is a risky arbitrage for agent j.
To see this, note that yj = limktkxkj with tk= k+11 and xkj = ωj+ (k + 1)yj
and xk
j ∈Pbj(ωj, x−j) since yj ∈ O+(Pbj(ωj, x−j)). We have then
n (xk j, xk−j) o k= {(ωj+ (k + 1)yj, x−j)}k⊂ X, and yj = limktkxkj, wheretk↓ 0, andwheref orallk, uj(xkj, xk−j) = uj(ωj+ (k + 1)yj, x−j) ≥ uj(ωj, x−j).
Example: Part 2, Recession Cones and Risky Arbitrage. Continuing our asset market example, let
sj(+) := limc→∞ dUj(c)dc
and
sj(−) := limc→−∞ dUj(c)dc
denote the asymptotic values of derivatives of the jth agent’s utility function Uj(·) in the positive and negative directions respectively and add to our list
of assumptions the following assumption:
(a-6) for each agent j the ratio of the asymptotic derivatives sj := sj(+) sj(−) is
zero.
The ratio sj is an asymptotic measure of risk tolerance. It inversely
measures the concavity of Uj(·) as c goes from −∞ to ∞. We adopt the
convention that sj = 0 when sj(−) = ∞. Note that under assumption (a-1)
(see Example, Part 1), 0 ≤ sj ≤ 1 for all agents j. If sj = 1 then the agent
is risk neutral and thus has the highest level of asymptotic risk tolerance. If sj <1 the agent is risk averse and therefore has a lower level of asymptotic risk tolerance. In particular, if sj = 0 then the agent has the lowest level of
asymptotic risk tolerance. It is easy to verify that sj = 0 for all constant
absolute risk aversion utility functions.
By Lemma 5.2 in Page (1987), if sj = 0, then
O+(Pbj(ωj, x−j)) = K+(x−j),
where, as in Example: Part 1, K+(x−j) is the positive dual cone of K(x−j).5 5Recall that K(x
−j) is the convex cone generated by the support S[µj(·|x−j)] of the
We now have our main result characterizing risky arbitrage. (Characterization of Risky Arbitrage)
Let (Xj, ωj, uj(·, ·))nj=1 be an economy with trading externalities
satisfy-ing assumptions [A-1]-[A-3]. The followsatisfy-ing statements are equivalent: 1. A vector of net trades yj ∈ RL is a risky arbitrage for agent j.
2. There exists a sequence n(xk j, xk−j) o k ⊂ X such that yj ∈ 0+ ³ limPbj(ωj, xk−j) ´ .
The limit, limPbj(ωj, xk−j), of the sequence of closed sets
n b
Pj(ωj, xk−j)
o
k
in part 2 of Theorem 1 is taken with respect to Painleve-Kuratowski con-vergence (see, for example section B.II p. 15 in Hildenbrand (1974) for definitions and properties). Before we prove Theorem 1, we provide the following Lemma.
Let (Xj, ωj, uj(·, ·))nj=1 be an economy with trading externalities
satisfy-ing assumptions [A-1]-[A-3]. Letn(xk j, xk−j)
o
k⊂ X be a sequence such that
for all j and k, xk
j ∈ Pbj(ωj, xk−j). Also let
n
tko
k be a sequence of positive
real numbers with tk ↓ 0. If (y
1, . . . , yn) is a cluster point of the sequence
n
(tkxk
1, . . . , tkxkn)
o
k,then there exists a subsequence
n
(tk′
xk1′, . . . , tk′xkn′)o
k′
such that for all j, yj ∈ 0+
³ limPbj(ωj, xk ′ −j) ´ .
Proof: (Lemma 3) Without loss of generality, assume that (y1, . . . , yn) = lim
k (t kxk
1, . . . , tkxkn).
From Hildenbrand (1974), Proposition 1, p. 16, there exists a converging subsequencen³Pb1(ω1, xk ′ −1), . . . ,Pbn(ωn, xk ′ −n) ´o k′of n³ b P1(ω1, xk−1), . . . ,Pbn(ωn, xk−n) ´o k.
Observe that for all j, limPbj(ωj, xk ′
−j) is convex (see Danzig, Folkman, and
Shapiro (1967), p. 521) and nonempty since it contains ωj. Also note that
(y1, . . . , yn) = limk′(tk ′
xk1′, . . . , tk′xkn′). Now let x∗j ∈ limPbj(ωj, xk
′
−j) and let t be any positive number. By
the definition of limPbj(ωj, xk ′
−j), there exists a sequence
n
x∗kj ′o
k′ such that
x∗kj ′ → x∗
j, as k′ → ∞, and for all k′, x∗k ′ j ∈Pbj(ωj, xk ′ −j). Since Pbj(ωj, xk ′ −j)
is convex for k′ large enough so that tk′
t≤ 1, (1 − tk′ t)x∗k′ j + tk ′ txkj′ ∈Pbj(ωj, xk ′ −j).
But (1 − tk′ t)x∗k′ j + tk ′ txk′ j → x∗j+ tyj ∈ limPbj(ωj, xk ′ −j). Thus, yj ∈ 0+ ³ limPbj(ωj, xk ′ −j) ´ .
Proof: (Theorem 3) (1) ⇒ (2). Let yj be a risky arbitrage for agent j
and letn(xk j, xk−j) o k⊂ X be such that f orallk, uj(xkj, xk−j) ≥ uj(ωj, xk−j), and yj = limktkxkjf ortk↓ 0. Then eithern°°°xkj°°°o k is bounded and yj = 0 or n° ° °xkj°°°o k is unbounded and
from the Lemma, yj ∈ 0+
³
limPbj(ωj, xk ′ −j)
´
for some subsequencen(xk′ j , xk ′ −j) o k′. (2) ⇒ (1). Conversely, let yj ∈ 0+ ³ limPbj(ωj, xk−j) ´
for some sequence
n
(xkj, xk−j)o
k⊂ X.
Let {λm}
m be a sequence of real numbers such that λm↑ ∞. Since
yj ∈ 0+
³
limPbj(ωj, xk−j)
´
,
we have ωj+ λm yj ∈ limPbj(ωj, xk−j) for all m. Let ε > 0. For each m there
exists km and xkmj ∈Pbj(ωj, xkm−j) such that
° ° °ωj+ λmyj− xkmj ° ° °< ε. This implies that °
° ° ° ωj λm + yj− xkmj λm ° ° ° °< λεm.
Letting m → ∞, we conclude that xkmj
λm → yj. Because xkmj ∈ Pbj(ωj, xkm−j)
for all m and because λ1m → 0, yj is a risky arbitrage for agent j.
(Closedness of the set of Risky Arbitrages)
Let (Xj, ωj, uj(·, ·))nj=1 be an economy with trading externalities
sat-isfying assumptions [A-1]-[A-3]. Then, for each agent j, the set of risky arbitrages, Rj,is closed.
Proof: Let {yν}
ν ⊂ Rj be a sequence of arbitrages for the jth agent
such that
We want to show that y ∈ Rj. By our characterization of risky
arbi-trage, we have for each ν, a sequence n(xk,νj , xk,ν−j)o
k ⊂ X such that y ν ∈
0+³limkPbj(ωj, xk,ν−j)
´
.Let ε > 0 and let {λm}m be a sequence of real
num-bers such that λm ↑ ∞. For all m and ν there exists a positive integer k(m, ν)
such that (i)°°°ωj+ λmyν − xk(m,ν),νj ° ° °≤ ε and (ii)xk(m,ν),νj ∈Pbj(ωj, xk(m,ν),ν−j ).
From (i) it follows that
° ° ° ° ° ωj λm + yν − xk(m,ν),νj λm ° ° ° ° °≤ ε λm. Therefore, ° ° ° ° ° xk(m,ν),νj λm ° ° ° ° °≤ ε λm + ° ° °λmωj ° ° °+ kyνk , and hence ( xk(m,ν),νj λm ) (m,ν)
is bounded. In particular, the sequence
(
xk(n,n),nj λn
)
n
is bounded. Let zj be a cluster point of this sequence. Then zj is a risky
arbitrage and zj = y.
(The No-Risky-Arbitrage Price Condition, NRAP):
(1) p ∈ RLis a NRAP price for agent j if hp, yji > 0 for all nonzero risky
arbitrages yj ∈ Rj\{0}.
(2) Let Sj denote the jth agent’s set of NRAP prices. The economy
(Xj, ωj, uj(·, ·))nj=1 with trading externalities satisfies NRAP if
∩jSj 6= ∅.
Note that the set of NRAP prices Sj is a convex cone. More
impor-tantly, note that any price vector p ∈ ∩jSj assigns a positive value to the
risky arbitrages of any agent. Thus, if p is a no-risky-arbitrage price, then each agent’s demand correspondence is nonempty at p no matter what con-sumption vectors are chosen by other agents (see section 6 below on viable prices, and in particular, see part 1 of Theorem 6.1).
If there exists a pointed closed convex cone C ⊂ RL such that each
agent’s set of risky arbitrages, Rj, is contained in C, then NRAP is
sat-isfied.6 In particular, by classical separation arguments, for C a pointed closed convex cone there exists a nonzero vector p ∈ RL such that hp, yi > 0
for all nonzero y ∈ C - and thus, hp, yji > 0 for all nonzero risky arbitrages
yj ∈ Rj. Conversely, since the risky arbitrage sets, Rj, are cones, if NRAP
is satisfied then given prices p contained in ∩jSj, there exists α > 0 such
that for all j, Rj is contained in the pointed convex cone C given by
C =ny∈ RL: hp, yi > α kyko.
Example: Part 3, The Existence of a Closed Pointed Cone Con-taining All Risky Arbitrages.
Assume that
(a-7) each agent j has conditional probability beliefs, µj(·|x−j), concerning
asset returns such that for some closed convex cone, Kj, with
non-empty interior
Kj = K(x−j)f orall(xj, x−j) ∈ X,
where again K(x−j) is the convex cone generated by the support
S[µj(·|x−j)] of µj(·|x−j).
It is important to note that the invariance of the cones K(x−j) with
respect to x−j (i.e., with respect to the trades of other agents) does not
imply that conditional probability beliefs are invariant with respect to x−j.
Moreover, nonemptiness of the interior of Kj implies that no asset returns
are perfectly correlated.
In light of Example: Part 2, we can conclude that if assumptions (a-1)-(a-6) are satisfied and if assumption (a-7) holds, then for all (xj, x−j) ∈ X
O+(Pbj(ωj, x−j)) = Kj+,
where Kj+ is the positive dual cone of Kj. Moreover, under (a-1)-(a-7), for
each agent j the set of risky arbitrages Rj is equal to Kj+.
By Proposition 3 in Page (1996), the jth agent’s set of NRAP prices, Sj,
is equal to the interior of Kj (denoted intKj), and thus, NRAP is satisfied
if and only if
∩jintKj 6= ∅
6Some authors take ”pointed” to mean only that the cone contains the origin. We
(i.e., a price vector p is a vector of no-risky-arbitrage prices if and only if p∈ ∩jintKj). Finally, by Proposition 5 in Page (1996), under assumptions
(a-1)-(a-7), ∩jintKj 6= ∅ if and only if n
X
j=1
yj = 0withyj ∈ Rjf oralljimpliesthatyj = 0f orallj.(∗)
Under (a-1)-(a-7) it is easy to show that condition (*), the no-mutually-compatible-arbitrages condition, holds if and only if there is a pointed closed convex cone C ⊂ RL such that each agent’s set of risky arbitrages, R
j, is
contained in C.
One of the main implications of NRAP is compactness of the set of rational allocations. This implication is a key ingredient in our proof of existence of a competitive equilibrium.
(NRAP implies compactness of rational allocations):
Let (Xj, ωj, uj(·, ·))nj=1 be an economy with trading externalities
satisfy-ing assumptions [A-1]-[A-3]. If the economy satisfies NRAP then the set of rational allocations, A, is compact.
Proof: Since A is closed, we have just to prove that A is bounded. Suppose not. Then there is a sequence n(xk
1, . . . , xkn) o k ⊂ A such that P j ° ° °xkj ° ° ° → ∞ as k → ∞. Letting tk := P 1 jkx k jk
, we have for some subse-quencen(xk′ 1 , . . . , xk ′ n) o k′, (tk′ xk1′, . . . , tk′xkn′) → (y1, . . . , yn) with P jkyjk = 1.
We have (y1, . . . , yn) 6= 0 and by definition, (y1, . . . , yn) is a risky arbitrage.
By NRAP, there exists a price vector p ∈ ∩jSj such that
hp, yji > 0f orj = 1, 2, . . . , n. Thus, X j hp, yji = * p,X j yj + >0. But now we have a contradiction because
P jtk ′ xkj′ =Pjtk ′ ωjf orallk andtheref ore P jtk ′ xkj′ →Pjyj = 0.
4
Existence of Equilibrium
4.1 Existence for Bounded Economies with Externalities
We begin by defining a k-bounded economy,
(Xkj, ωj, uj(·, ·))nj=1, (6)
In the k-bounded economy, the jth agent’s consumption set is
Xkj := Xj∩ Bk(ωj), (7)
where Bk(ωj) is a closed ball of radius k centered at the agent’s endowment,
ωj. Define Xk:= n Y j=1 Xkj.
The set of k-bounded rational allocations is given by Ak= {(x1, ..., xn) ∈ Xk:Pnj=1xj =Pnj=1ωj
andf oreachj, uj(xj, x−j) ≥ uj(ωj, x−j)}.
(8) By Theorem 3.6 above, if the original economy (Xj, ωj, uj(·, ·))nj=1
sat-isfies NRAP, then the set of rational allocations is compact. Thus, there exists some integer k∗ such that for all k ≥ k∗, A
k= A.
An equilibrium for the k-bounded economy, (Xkj, ωj, uj(·))nj=1,is an (n+
1)-tuple of vectors (xk
1, . . . , xkn, pk) such that
(i) (xk
1, . . . , xkn) ∈ Ak,(the allocation is feasible);
(ii) pk ∈ B\{0} (prices are in the unit ball and not all prices are zero);
and
(iii) for each j, (a) Dpk, xk j E =Dpk, ω j E
(budget constraints are satisfied), and (b) xk
j ∈ Bkj(pk, ωj) and Pkj(xkj, xk−j) ∩ Bkj(pk, ωj) = ∅ (i.e., xkj
Here,
Pkj(xkj, xk−j) := Pj(xkj, xk−j) ∩ Xkj,
and
Bkj(pk, ωj) := Bj(pk, ωj) ∩ Xkj.
We now have our main existence result for bounded economies. (Existence of an equilibrium for k-bounded economies)
Let (Xj, ωj, uj(·))nj=1 be an economy with trading externalities satisfying
assumptions [A-1]-[A-3] and let k∗ satisfy the condition that A
k∗ = A. Then
for all k ≥ k∗ the k-bounded economy,
(Xjk, ωj, uj(·, ·))nj=1, has an equilibrium, (xk 1, . . . , xkn, pk), with pk∈ Bu := n p∈ RL: kpk = 1o.
Proof: Because the original economy (Xj, ωj, uj(·, ·))nj=1 satisfies
lo-cal nonsatiation at rational allocations (i.e., assumption [A-3]), for all k larger than k∗ such that A
k = A for k ≥ k∗, the k-bounded economy
(Xjk, ωj, uj(·, ·))nj=1 also satisfies local nonsatiation at rational allocations.
Thus it follows from Florenzano (2003), chapter 2, that for k larger than k∗,
the k-bounded economy (Xjk, ωj, uj(·, ·))nj=1 has an equilibrium.
4.2 Existence for Unbounded Economies with Externalities
Our main existence result for unbounded economies with externalities is the following:
(Existence for unbounded economies with externalities)
Let (Xj, ωj, uj(·, ·))nj=1 be an economy with trading externalities
sat-isfying assumptions [A-1]-[A-3]. If the economy satisfies NRAP, then the economy has an equilibrium, (x1, . . . , xn, p), with
p∈ Bu :=
n
p∈ RL: kpk = 1o.
.
Proof: For each k larger than k∗ such that Ak = A for k ≥ k∗, the
k-bounded economy (Xjk, ωj, uj(·, ·))nj=1 has an equilibrium
(xk
Since A × Bu is compact, we can assume without loss of generality that
(xk1, . . . , xkn, pk) → (x1, . . . , xn, p) ∈ A × Bu.
Moreover, since for all j and k, Dpk, xkjE = Dpk, ωj
E
, we have for all j, hp, xji = hp, ωji .
Let uj(xj, x−j) > uj(xj, x−j). Then, for k > k∗ sufficiently large, xj ∈
Xjk and uj(xj, xk−j) > uj(xkj, xk−j) which implies that
D pk, x j E > Dpk, ω j E . Thus, in the limit hp, xji ≥ hp, ωji . Hence, (x1, . . . , xn, p) is a quasi-equilibrium.
Since for all j, ωj ∈ intXj (see [A-1]) and since utility functions are
contin-uous (see [A-2]), in fact, (x1, . . . , xn, p) is an equilibrium.
5
Necessary and Sufficient Conditions for
Exis-tence
We begin by introducing the following uniformity conditions: [A − 4]
(
If yj ∈ Rj\ {0} , then
f orall(xj, x−j) ∈ A, uj(xj+ yj, x−j) > uj(xj, x−j).
By assumption [A-4] all risky arbitrages are utility increasing provided that the starting point for the risky arbitrage is a rational allocation.
Now we have our main result on necessary and sufficient conditions for existence.
(NRAP ⇔ existence of equilibrium)
Let (Xj, ωj, uj(·, ·))nj=1 be an economy with trading externalities
satisfy-ing assumptions [A-1]-[A-4]. Then the followsatisfy-ing statements are equivalent: 1. (Xj, ωj, uj(·, ·))nj=1 satisfies NRAP.
2. (Xj, ωj, uj(·, ·))nj=1 has an equilibrium.
Proof: By Theorem 4.2, we know that (1) ⇒ (2). So we need only establish that (2) ⇒ (1). Let (x, p) be an equilibrium and for some agent j suppose that yj ∈ Rj\ {0} is a risky arbitrage. By [A-4], uj(xj+ yj, x−j) >
uj(xj, x−j). Because (x, p) is an equilibrium hp, xj+ yji > hp, ωji = hp, xji .
6
Viable Prices and Externalities
In this section we extend Kreps’ (1981) notion of viable prices to exchange economies with externalities and establish the relationship between NRAP and viable prices. To begin, consider the problem
maxnuj(xj, x−j) : xj ∈Pbj(ωj, x−j)and hp, xji ≤ hp, ωji
o
,
where x−j ∈ X−j is given. We say that price vector p is viable for agent
j if this problem has a solution for any x−j ∈ X−j. Thus, p is viable for
agent j if agent j′s demand correspondence is nonempty at p no matter
what consumption vector x−j ∈ X−j is chosen by other agents. Consider
now the following strengthening of assumption [A-4], [A − 4]∗
(
If yj ∈ Rj\ {0} , then
f orall(xj, x−j) ∈ X, uj(xj+ yj, x−j) > uj(xj, x−j).
By assumption [A-4]∗ all risky arbitrages are utility increasing starting at
any (xj, x−j) ∈ X.
(NRAP and viable prices)
Let (Xj, ωj, uj(·, ·))nj=1 be an economy with trading externalities
satisfy-ing assumptions [A-1]-[A-3]. Then the followsatisfy-ing statements are true: 1. If p is an NRAP price for agent j, then p is viable for agent j.
2. If assumption [A-4]∗ also holds, then if p is viable for agent j, then p
is an NRAP price for agent j.
Proof: (1) Since uj(·, ·) is continuous, it suffices to prove that the set
n
x∈ RL: x ∈Pb
j(ωj, x−j)and hp, xi ≤ hp, ωji
o
is bounded. If not, letnxko
k be an unbounded sequence which satisfies
xk∈Pbj(ωj, x−j)
and
D
p, xkE≤ hp, ωji f orallk.
Let y be a cluster point of the sequence
½
xk
kxkk
¾
k
.Then y is a risky arbitrage and hp, yi ≤ 0, a contradiction since p is an NRA price vector for agent j.
(2) Conversely, let p be viable and assume [A-4]∗ holds. Let x solve the
problem
maxnuj(xj, x−j) : xj ∈Pbj(ωj, x−j)and hp, xji ≤ hp, ωji
o
, and suppose y 6= 0 is a risky arbitrage. By [A-4]∗
uj(x + y, x−j) > uj(x, x−j).
We have
hp, ωj+ yi ≥ hp, x + yi
and
hp, x + yi > hp, ωji implies hp, yi > 0.
By Theorem 6.1, if the economy satisfies [A-1]-[A-3] then the NRAP condition guarantees the existence of a nonempty set of viable prices for the economy (i.e., for all agents), and thus, guarantees the existence of demand functions over the set of viable prices. In addition, by Theorem 6.1, if all risky arbitrages are utility increasing starting at any (xj, x−j) ∈ X
(i.e., if [A-4]∗ holds), then the existence of demand functions guarantees the
existence no-risky-arbitrage prices.
Example: Part 4, The Uniformity Conditions [A-4] and [A-4]*: Under assumptions (a-1)-(a-7), it follows from Lemma 3 in Page (1996) that each risky arbitrage yj ∈ Rj\{0} is such that
uj(xj+ yj, x−j) > uj(xj, x−j)f orall(xj, x−j) ∈ X.
Thus, in our asset market example, if assumptions (a-1)-(a-7) hold, then all risky arbitrages yj ∈ Rj\{0} are utility increasing starting at any (xj, x−j) ∈
X (i.e., the uniformity assumption [A-4]* holds - and thus [A-4] holds as well).
7
Conclusions
Externalities are a pervasive feature of economics and, not surprisingly, the subject of ongoing interest in general equilibrium models (see, for example, Florenzano (2003), Bonnisseau (1997), Bonnisseau and M´edecin (2001)). Our research contributes to this for a class of models which we feel is of interest and importance – situations where agents may be affected by both prices and trading volume, an indicator of what other agents are doing. Our condition, NRAP forges a link between trading volume and asset prices in markets where arbitrage is possible.
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